Association and Causation in a Wellness Survey
A school wellness committee surveys 300 students about after-school program participation and chronic absence, defined as more than 10 days absent. Results: 120 students participate in the program, and 48 of them are chronically absent. Of the 180 nonparticipants, 90 are chronically absent. A committee member reports: 'The program cuts chronic absence from 50% to 40%, a 10-point reduction,' and recommends expansion. Another member notes that students who join may already be more engaged. The principal asks for a statistical appraisal before allocating funds. The 300 students were not randomly assigned; some volunteered and some were referred by counselors. The committee has the raw counts and access to a randomization app but no additional demographic data.
1. Compute and interpret the conditional probabilities of chronic absence for participants and nonparticipants. What association does the table show?
Read the worked answer
P(chronic absence | program) = 48/120 = 0.40. P(chronic absence | no program) = 90/180 = 0.50. The risk difference is 0.40 − 0.50 = −0.10, and the risk ratio is 0.40/0.50 = 0.80. The table shows an association: program participants have a 10-percentage-point lower observed chronic-absence rate, and their risk is 20% lower relative to nonparticipants. This is a descriptive statement about the sample, not an estimate of a causal effect, because participation was not randomly assigned and the groups may differ in prior attendance, grade level, motivation, or other factors.
2. Explain two plausible alternative explanations for the association and how randomization would address them.
Read the worked answer
Alternative explanation 1: self-selection. Students who choose the program may already be more engaged or have better transportation and family support, so they would have lower absence even without the program. Alternative explanation 2: referral patterns. If counselors refer students with emerging attendance problems, the program group could be higher-risk than it appears; the observed lower rate might then underestimate a benefit, or the groups might differ in ways that obscure the effect. Random assignment would break the link between these confounders and program participation; over many randomizations, the groups would be similar on average, so a difference in absence could be attributed more credibly to the program.
3. The committee wants a 10-minute activity to help students interpret the 10-point difference. Design it and state the conclusion students should reach.
Read the worked answer
Activity: give pairs the two-way table and ask them to compute the two conditional probabilities, then draw two bars (40% vs 50%). Next, have them simulate a 'no-effect' world by shuffling 300 cards labeled with the 138 chronic absences and 162 non-absences into two groups of 120 and 180, repeatedly, and record the difference in rates. If the observed 10-point difference is unusual relative to the shuffle distribution, that supports an association; if it is common, the difference could be noise. Either way, students should conclude that the survey alone does not establish that the program causes lower absence; a randomized or well-controlled comparison is needed before funding expansion.